SIGMA 3 (2007), 068, 12 pages      arXiv:0705.2671      https://doi.org/10.3842/SIGMA.2007.068
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Hidden Symmetries of Stochastic Models

Boyka Aneva
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received November 23, 2006, in final form May 04, 2007; Published online May 18, 2007

Abstract
In the matrix product states approach to n species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process. The quadratic algebra defines a noncommutative space with a SU_q(n) quantum group action as its symmetry. Boundary processes amount to the appearance of parameter dependent linear terms in the algebraic relations and lead to a reduction of the SU_q(n) symmetry. We argue that the boundary operators of the asymmetric simple exclusion process generate a tridiagonal algebra whose irriducible representations are expressed in terms of the Askey-Wilson polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary problem and allows to solve the model exactly.

Key words: stohastic models; tridiagonal algebra; Askey-Wilson polynomials.

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References

1. Schuetz G.M., Phase transitions and critical phenomena, Vol. 19, Academic Press, London, 2000.
2. Schreckenberg M., Schadschneider A., Nagel K., Ito N., Discrete stochastic models for traffic flow, Phys. Rev. E 51 (1995), 2939-2949, cond-mat/9412045.
3. Macdonald J.T., Gibbs J.H., Pipkin A.C., Kinetics of biopolymerization on nucleic acid templates, Biopolymers 6 (1968), 1-25.
4. Krug J., Spohn H., Kinetic roughening of growing surfaces, in Solids Far from Equilibrium, Editor C. Godreche, Cambridge University Press, Cambridge, 1991, 412-525.
5. Sandow S., Schuetz G.M., On U_q(SU(2))-symmetric driven diffusion, Europhys. Lett. 26 (1994), 7-13, cond-mat/9307027.
6. Derrida B., Evans M.R., Hakim V., Pasquier V., Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A: Math. Gen. 26 (1993), 1493-1517.
7. Derrida B., An exactly soluble non-equilibrium system: the asymmetric simple exclusion process, Phys. Rep. 301 (1998), 65-83 (and references therein).
8. Schuetz G.M., Domany E., Phase transitions in an exactly soluble one-dimensional asymmetric exclusion model, J. Stat. Phys. 72 (1993), 277-296, cond-mat/9303038.
9. Krug J., Boundary-induced phase transitions in driven diffusive systems, Phys. Rev. Lett. 67 (1991), 1882-1885.
10. Isaev A., Pyatov P., Rittenberg V., Diffusion algebras, J. Phys. A: Math. Gen. 34 (2001), 5815-5834, cond-mat/0103603.
11. Aneva B., The noncommutative space of stochastic diffusion systems, J. Phys. A: Math. Gen. 35 (2002), 859-877.
12. Essler F.H.L., Rittenberg V., Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries, J. Phys. A: Math. Gen. 29 (1996), 3375-3407, cond-mat/9506131.
13. Arndt P.F., Heinzel T., Rittenberg V., Stochastic models on a ring and quadratic algebras. The three-species diffusion problem, J. Phys. A: Math. Gen. 31 (1998), 833-843, cond-mat/9703182.
14. Fairlie D., Zachos C., Multiparameter associative generalizations of canonical commutation relations and quantized planes, Phys. Lett. B 256 (1991), 43-49.
15. Wess J., Zumino B., Covariant differential calculus on the quantum hyperplane, Nucl. Phys. B, Proc. Suppl. 18 (1990), 302-312.
16. Blythe R.A., Evans M.R., Colaiori F., Essler F.H.L., Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra, J. Phys. A: Math. Gen. 33 (2000), 2313-2332.
17. Sandow S., Partially asymmetric exclusion process with open boundaries, Phys. Rev. E 50 (1994), 2660-2667, cond-mat/9405073.
18. Stinchcombe R.B., Schuetz G.M., Application of operator algebras to stochastic dynamics and the Heisenberg chain, Phys. Rev. Lett. 75 (1995), 140-143.
19. Arik M., Coon D.D., Hilbert spaces of analytic functions and generalized coherent states, J. Math. Phys. 17 (1976), 524-527.
20. Kulish P.P., Damaskinsky E.V., On the q oscillator and the quantum algebra su_q(1,1), J. Phys. A: Math. Gen. 23 (1990), L415-L419.
21. Chaichian M., Kulish P.P., Quantum Lie superalgebras and q-oscillators, Phys. Lett. B 234 (1990), 72-80.
22. Sasamoto T., One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach, J. Phys. A: Math. Gen. 32 (1999), 7109-7131.
23. Uchiyama M., Sasamoto T., Wadati M., Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials, J. Phys. A: Math. Gen. 37 (2004), 4985-5002, cond-mat/0312457.
24. Zhedanov A.S., Hidden symmetry of Askey-Wilson polynomials, Teoret. Mat. Fiz. 89 (1991), 190-204.
25. Granovskii Y.A., Lutsenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
26. Terwilliger P., An algebraic approach to the Askey scheme of orthogonal polynomials, Lecture Notes in Mathematics, Vol. 1883, Editors F. Marcellan and W.V. Assche, Springer, Berlin, 2006, 225-330.
27. Terwilliger P., Two relations that generalize the q-Serre relations and the Dolan-Grady relations, Proceedings of the Nagoya 1999 International Workshop on Physics and Combinatorics, Editors A.N. Kirillov, A. Tsuchiya and H. Umemura, World Sci. Pub., River Edge, NJ, 2001, 377-398, math.QA/0307016.
28. Dolan L., Grady M., Conserved charges from self-duality, Phys. Rev. D25 (1982), 1587-1604.
29. Askey R.A., Wilson J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. (1985), no. 319, 55 pages.
30. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and Its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990.